Exercise 25, § 4

One says that a real function $f$ defined on $[0, +\infty[$ is superadditive if one has $f(x+y) \geq f(x) + f(y)$ for $x \geq 0, y \geq 0$, and if $f(0) = 0$. a) Give examples of discontinuous superadditive functions.

b) Show that every convex function $f$ on $[0, +\infty[$ such that $f(0) = 0$ is superadditive.

c) If $f_1$ and $f_2$ are superadditive then so is $\inf(f_1, f_2)$; using this, exhibit examples of nonconvex continuous superadditive functions.

d) If $f$ is continuous and $\geqslant 0$ on an interval $[0, a]$ ($a > 0$), such that $f(0) = 0$ and $f(x/n) \leqslant f(x)/n$ for each integer $n \geqslant 1$, show that $f$ has a right derivative at the point 0 (argue by contradiction). In particular, every continuous superadditive function which is $\geqslant 0$ admits a right derivative at the point 0.